July  2011, 10(4): 1097-1109. doi: 10.3934/cpaa.2011.10.1097

The Cauchy-Kovalevskaya extension theorem in discrete Clifford analysis

1. 

Cli ord Research Group, Faculty of Engineering, Ghent University, Galglaan 2, 9000, Gent, Belgium

2. 

Clifford Research Group, Department of Mathematical Analysis, Faculty of Engineering, Ghent University, Galglaan 2, 9000 Gent, Belgium

3. 

Clifford Research Group, Faculty of Sciences, Ghent University, Galglaan 2, 9000 Gent

Received  October 2010 Revised  January 2011 Published  April 2011

Discrete Clifford analysis is a higher dimensional discrete function theory based on skew Weyl relations. It is centered around the study of Clifford algebra valued null solutions, called discrete monogenic functions, of a discrete Dirac operator, i.e. a first order, Clifford vector valued difference operator. In this paper, we establish a Cauchy-Kovalevskaya extension theorem for discrete monogenic functions defined on the standard $Z^m$ grid. Based on this extension principle, discrete Fueter polynomials, forming a basis of the space of discrete spherical monogenics, i.e. homogeneous discrete monogenic polynomials, are introduced. As an illustrative example we moreover explicitly construct the Cauchy-Kovalevskaya extension of the discrete delta function. These results are then generalized for a grid with variable mesh width $h$.
Citation: Hilde De Ridder, Hennie De Schepper, Frank Sommen. The Cauchy-Kovalevskaya extension theorem in discrete Clifford analysis. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1097-1109. doi: 10.3934/cpaa.2011.10.1097
References:
[1]

F. Brackx, R. Delanghe and F. Sommen, "Clifford Analysis,", Research Notes in Mathematics, 76 (1982). Google Scholar

[2]

F. Brackx, H. De Schepper, F. Sommen and L. Van de Voorde, Discrete Clifford analysis: an overview,, Cubo, 11 (2009), 55. Google Scholar

[3]

F. Brackx, H. De Schepper, F. Sommen and L. Van de Voorde, Discrete Clifford analysis: a germ of function theory,, In: I. Sabadini, (2009), 37. Google Scholar

[4]

R. Delanghe, F. Sommen and V. Souček, "Clifford Algebra and Spinor-valued Functions - A Function Theory for the Dirac Operator,", Kluwer Academic Publishers, (1992). Google Scholar

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A. Cauchy, Oeuvres completes,, S\'erie 1, (): 1882. Google Scholar

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R. Cooke, The Cauchy-Kovalevskaya Theorem, (preprint, (). Google Scholar

[7]

H. De Ridder, H. De Schepper, F. Sommen and U. Kähler, Discrete function theory based on skew Weyl relations,, Proc. Amer. Math. Soc., 138 (2010), 3241. Google Scholar

[8]

H. De Ridder, H. De Schepper and F. Sommen, Fueter polynomials in discrete Clifford analysis,, (submitted)., (). Google Scholar

[9]

N. Faustino, U. Kähler and F. Sommen, Discrete Dirac operators in Clifford analysis,, Adv. Appl. Cliff. Alg., 17 (2007), 451. Google Scholar

[10]

J. Gilbert and M. Murray, "Clifford Algebra and Dirac Operators in Harmonic Analysis,", Cambridge University Press, (1991). Google Scholar

[11]

K. Gürlebeck and W. Sprössig, "Quaternionic and Clifford Calculus for Physicists and Engineers,", J. Wiley & Sons, (1997). Google Scholar

[12]

S. Kowalevsky, Zur Theorie der partiellen Differentialgleichung,, J. f\, 80 (1875), 1. Google Scholar

show all references

References:
[1]

F. Brackx, R. Delanghe and F. Sommen, "Clifford Analysis,", Research Notes in Mathematics, 76 (1982). Google Scholar

[2]

F. Brackx, H. De Schepper, F. Sommen and L. Van de Voorde, Discrete Clifford analysis: an overview,, Cubo, 11 (2009), 55. Google Scholar

[3]

F. Brackx, H. De Schepper, F. Sommen and L. Van de Voorde, Discrete Clifford analysis: a germ of function theory,, In: I. Sabadini, (2009), 37. Google Scholar

[4]

R. Delanghe, F. Sommen and V. Souček, "Clifford Algebra and Spinor-valued Functions - A Function Theory for the Dirac Operator,", Kluwer Academic Publishers, (1992). Google Scholar

[5]

A. Cauchy, Oeuvres completes,, S\'erie 1, (): 1882. Google Scholar

[6]

R. Cooke, The Cauchy-Kovalevskaya Theorem, (preprint, (). Google Scholar

[7]

H. De Ridder, H. De Schepper, F. Sommen and U. Kähler, Discrete function theory based on skew Weyl relations,, Proc. Amer. Math. Soc., 138 (2010), 3241. Google Scholar

[8]

H. De Ridder, H. De Schepper and F. Sommen, Fueter polynomials in discrete Clifford analysis,, (submitted)., (). Google Scholar

[9]

N. Faustino, U. Kähler and F. Sommen, Discrete Dirac operators in Clifford analysis,, Adv. Appl. Cliff. Alg., 17 (2007), 451. Google Scholar

[10]

J. Gilbert and M. Murray, "Clifford Algebra and Dirac Operators in Harmonic Analysis,", Cambridge University Press, (1991). Google Scholar

[11]

K. Gürlebeck and W. Sprössig, "Quaternionic and Clifford Calculus for Physicists and Engineers,", J. Wiley & Sons, (1997). Google Scholar

[12]

S. Kowalevsky, Zur Theorie der partiellen Differentialgleichung,, J. f\, 80 (1875), 1. Google Scholar

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